We have $$e^{i\lambda x}\cdot e^{i\mu x}=e^{i(\lambda+\mu) x}.\ (1)$$ More generally, consider the Clebsch–Gordan coefficients $c_{\lambda,\mu}^\nu$ defined by $$\pi_\lambda\otimes\pi_\mu=\sum_\nu c_{\lambda,\mu}^\nu \pi_{\nu}\ $$ where $\pi_\alpha$ stands for the irreducible representation of a compact Lie group of spectral parameter $\alpha$. On the circle group, because of (1), we have $$c_{\lambda,\mu}^\nu=0\text{ except when }\nu=\lambda+\mu.$$ My question is, in general, are there any simple vanishing results like the above or estimates such as $$c_{\lambda,\mu}^\nu=0\text{ for }|\nu-(\lambda+\mu)|\geq C?$$